The Laplacian flow is an evolution equation of closed G2-structures arising as the gradient flow of the so-called Hitchin volume functional. In this talk, we shall consider the flow of those G2 structures admitting S1 symmetry and derive explicitly the evolution equations of the SU(3)-structure on the quotient manifold together with a connection 1-form. We describe these equations in a couple of examples and mention some partial results of ongoing work.